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Properties

Label 567.2.bd.a.17.10
Level $567$
Weight $2$
Character 567.17
Analytic conductor $4.528$
Analytic rank $0$
Dimension $132$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(17,567)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("567.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([11, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.bd (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 17.10
Character \(\chi\) \(=\) 567.17
Dual form 567.2.bd.a.467.10

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.245063 + 0.0432112i) q^{2} +(-1.82120 + 0.662861i) q^{4} +(-1.99870 + 0.727467i) q^{5} +(0.302346 - 2.62842i) q^{7} +(0.848674 - 0.489982i) q^{8} +(0.458373 - 0.264642i) q^{10} +(0.489168 - 1.34398i) q^{11} +(1.30709 + 3.59121i) q^{13} +(0.0394834 + 0.657193i) q^{14} +(2.78250 - 2.33479i) q^{16} +(-0.109097 - 0.188961i) q^{17} +(2.56992 + 1.48374i) q^{19} +(3.15782 - 2.64972i) q^{20} +(-0.0618021 + 0.350497i) q^{22} +(7.61556 + 1.34283i) q^{23} +(-0.364628 + 0.305959i) q^{25} +(-0.475500 - 0.823590i) q^{26} +(1.19165 + 4.98728i) q^{28} +(1.78349 - 4.90009i) q^{29} +(1.32486 + 3.64001i) q^{31} +(-1.84082 + 2.19380i) q^{32} +(0.0349009 + 0.0415932i) q^{34} +(1.30779 + 5.47337i) q^{35} +6.01251 q^{37} +(-0.693906 - 0.252561i) q^{38} +(-1.33980 + 1.59671i) q^{40} +(10.0627 - 3.66253i) q^{41} +(1.29579 + 7.34878i) q^{43} +2.77190i q^{44} -1.92432 q^{46} +(5.51793 + 2.00836i) q^{47} +(-6.81717 - 1.58938i) q^{49} +(0.0761360 - 0.0907353i) q^{50} +(-4.76094 - 5.67387i) q^{52} +(-12.1036 - 6.98800i) q^{53} +3.04206i q^{55} +(-1.03129 - 2.37882i) q^{56} +(-0.225328 + 1.27790i) q^{58} +(1.21070 + 1.01590i) q^{59} +(1.45500 - 3.99757i) q^{61} +(-0.481963 - 0.834784i) q^{62} +(-3.27598 + 5.67416i) q^{64} +(-5.22497 - 6.22688i) q^{65} +(2.59043 - 14.6911i) q^{67} +(0.323942 + 0.271820i) q^{68} +(-0.557002 - 1.28481i) q^{70} +(4.29452 + 2.47944i) q^{71} -2.46806i q^{73} +(-1.47344 + 0.259808i) q^{74} +(-5.66384 - 0.998688i) q^{76} +(-3.38464 - 1.69209i) q^{77} +(-0.675969 - 3.83361i) q^{79} +(-3.86290 + 6.69073i) q^{80} +(-2.30773 + 1.33237i) q^{82} +(0.141502 + 0.0515026i) q^{83} +(0.355515 + 0.298313i) q^{85} +(-0.635099 - 1.74492i) q^{86} +(-0.243381 - 1.38028i) q^{88} +(-3.40271 + 5.89366i) q^{89} +(9.83439 - 2.34980i) q^{91} +(-14.7595 + 2.60251i) q^{92} +(-1.43902 - 0.253739i) q^{94} +(-6.21587 - 1.09603i) q^{95} +(12.2644 - 2.16255i) q^{97} +(1.73932 + 0.0949204i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.245063 + 0.0432112i −0.173286 + 0.0305549i −0.259618 − 0.965711i \(-0.583597\pi\)
0.0863320 + 0.996266i \(0.472485\pi\)
\(3\) 0 0
\(4\) −1.82120 + 0.662861i −0.910598 + 0.331431i
\(5\) −1.99870 + 0.727467i −0.893846 + 0.325333i −0.747784 − 0.663942i \(-0.768882\pi\)
−0.146062 + 0.989275i \(0.546660\pi\)
\(6\) 0 0
\(7\) 0.302346 − 2.62842i 0.114276 − 0.993449i
\(8\) 0.848674 − 0.489982i 0.300052 − 0.173235i
\(9\) 0 0
\(10\) 0.458373 − 0.264642i 0.144950 − 0.0836870i
\(11\) 0.489168 − 1.34398i 0.147490 − 0.405225i −0.843845 − 0.536588i \(-0.819713\pi\)
0.991334 + 0.131363i \(0.0419353\pi\)
\(12\) 0 0
\(13\) 1.30709 + 3.59121i 0.362522 + 0.996021i 0.978135 + 0.207972i \(0.0666864\pi\)
−0.615613 + 0.788049i \(0.711091\pi\)
\(14\) 0.0394834 + 0.657193i 0.0105524 + 0.175642i
\(15\) 0 0
\(16\) 2.78250 − 2.33479i 0.695625 − 0.583699i
\(17\) −0.109097 − 0.188961i −0.0264599 − 0.0458299i 0.852492 − 0.522740i \(-0.175090\pi\)
−0.878952 + 0.476910i \(0.841757\pi\)
\(18\) 0 0
\(19\) 2.56992 + 1.48374i 0.589579 + 0.340394i 0.764931 − 0.644112i \(-0.222773\pi\)
−0.175352 + 0.984506i \(0.556106\pi\)
\(20\) 3.15782 − 2.64972i 0.706109 − 0.592496i
\(21\) 0 0
\(22\) −0.0618021 + 0.350497i −0.0131762 + 0.0747262i
\(23\) 7.61556 + 1.34283i 1.58795 + 0.279999i 0.896709 − 0.442621i \(-0.145951\pi\)
0.691246 + 0.722620i \(0.257062\pi\)
\(24\) 0 0
\(25\) −0.364628 + 0.305959i −0.0729256 + 0.0611919i
\(26\) −0.475500 − 0.823590i −0.0932532 − 0.161519i
\(27\) 0 0
\(28\) 1.19165 + 4.98728i 0.225200 + 0.942508i
\(29\) 1.78349 − 4.90009i 0.331185 − 0.909924i −0.656619 − 0.754223i \(-0.728014\pi\)
0.987804 − 0.155702i \(-0.0497638\pi\)
\(30\) 0 0
\(31\) 1.32486 + 3.64001i 0.237951 + 0.653766i 0.999981 + 0.00622334i \(0.00198096\pi\)
−0.762029 + 0.647543i \(0.775797\pi\)
\(32\) −1.84082 + 2.19380i −0.325413 + 0.387813i
\(33\) 0 0
\(34\) 0.0349009 + 0.0415932i 0.00598545 + 0.00713318i
\(35\) 1.30779 + 5.47337i 0.221057 + 0.925168i
\(36\) 0 0
\(37\) 6.01251 0.988450 0.494225 − 0.869334i \(-0.335452\pi\)
0.494225 + 0.869334i \(0.335452\pi\)
\(38\) −0.693906 − 0.252561i −0.112566 − 0.0409708i
\(39\) 0 0
\(40\) −1.33980 + 1.59671i −0.211841 + 0.252462i
\(41\) 10.0627 − 3.66253i 1.57153 − 0.571990i 0.598190 − 0.801354i \(-0.295887\pi\)
0.973341 + 0.229364i \(0.0736646\pi\)
\(42\) 0 0
\(43\) 1.29579 + 7.34878i 0.197606 + 1.12068i 0.908659 + 0.417540i \(0.137108\pi\)
−0.711053 + 0.703139i \(0.751781\pi\)
\(44\) 2.77190i 0.417880i
\(45\) 0 0
\(46\) −1.92432 −0.283725
\(47\) 5.51793 + 2.00836i 0.804873 + 0.292950i 0.711504 − 0.702682i \(-0.248014\pi\)
0.0933687 + 0.995632i \(0.470236\pi\)
\(48\) 0 0
\(49\) −6.81717 − 1.58938i −0.973882 − 0.227055i
\(50\) 0.0761360 − 0.0907353i 0.0107673 − 0.0128319i
\(51\) 0 0
\(52\) −4.76094 − 5.67387i −0.660224 − 0.786824i
\(53\) −12.1036 − 6.98800i −1.66255 − 0.959875i −0.971490 − 0.237080i \(-0.923810\pi\)
−0.691063 − 0.722795i \(-0.742857\pi\)
\(54\) 0 0
\(55\) 3.04206i 0.410192i
\(56\) −1.03129 − 2.37882i −0.137811 − 0.317883i
\(57\) 0 0
\(58\) −0.225328 + 1.27790i −0.0295870 + 0.167796i
\(59\) 1.21070 + 1.01590i 0.157620 + 0.132259i 0.718187 − 0.695850i \(-0.244972\pi\)
−0.560567 + 0.828109i \(0.689417\pi\)
\(60\) 0 0
\(61\) 1.45500 − 3.99757i 0.186293 − 0.511837i −0.811026 − 0.585010i \(-0.801090\pi\)
0.997319 + 0.0731735i \(0.0233127\pi\)
\(62\) −0.481963 − 0.834784i −0.0612093 − 0.106018i
\(63\) 0 0
\(64\) −3.27598 + 5.67416i −0.409497 + 0.709270i
\(65\) −5.22497 − 6.22688i −0.648078 − 0.772349i
\(66\) 0 0
\(67\) 2.59043 − 14.6911i 0.316471 − 1.79480i −0.247376 − 0.968920i \(-0.579568\pi\)
0.563848 − 0.825879i \(-0.309321\pi\)
\(68\) 0.323942 + 0.271820i 0.0392838 + 0.0329630i
\(69\) 0 0
\(70\) −0.557002 − 1.28481i −0.0665745 − 0.153564i
\(71\) 4.29452 + 2.47944i 0.509666 + 0.294256i 0.732696 − 0.680556i \(-0.238262\pi\)
−0.223031 + 0.974811i \(0.571595\pi\)
\(72\) 0 0
\(73\) − 2.46806i − 0.288864i −0.989515 − 0.144432i \(-0.953864\pi\)
0.989515 − 0.144432i \(-0.0461355\pi\)
\(74\) −1.47344 + 0.259808i −0.171284 + 0.0302020i
\(75\) 0 0
\(76\) −5.66384 − 0.998688i −0.649687 − 0.114557i
\(77\) −3.38464 − 1.69209i −0.385716 − 0.192831i
\(78\) 0 0
\(79\) −0.675969 − 3.83361i −0.0760524 − 0.431315i −0.998931 − 0.0462239i \(-0.985281\pi\)
0.922879 − 0.385091i \(-0.125830\pi\)
\(80\) −3.86290 + 6.69073i −0.431885 + 0.748047i
\(81\) 0 0
\(82\) −2.30773 + 1.33237i −0.254847 + 0.147136i
\(83\) 0.141502 + 0.0515026i 0.0155319 + 0.00565315i 0.349774 − 0.936834i \(-0.386258\pi\)
−0.334243 + 0.942487i \(0.608480\pi\)
\(84\) 0 0
\(85\) 0.355515 + 0.298313i 0.0385610 + 0.0323566i
\(86\) −0.635099 − 1.74492i −0.0684845 − 0.188160i
\(87\) 0 0
\(88\) −0.243381 − 1.38028i −0.0259445 − 0.147139i
\(89\) −3.40271 + 5.89366i −0.360686 + 0.624727i −0.988074 − 0.153980i \(-0.950791\pi\)
0.627388 + 0.778707i \(0.284124\pi\)
\(90\) 0 0
\(91\) 9.83439 − 2.34980i 1.03092 − 0.246326i
\(92\) −14.7595 + 2.60251i −1.53879 + 0.271330i
\(93\) 0 0
\(94\) −1.43902 − 0.253739i −0.148424 − 0.0261712i
\(95\) −6.21587 − 1.09603i −0.637735 − 0.112450i
\(96\) 0 0
\(97\) 12.2644 − 2.16255i 1.24526 − 0.219573i 0.488092 − 0.872792i \(-0.337693\pi\)
0.757169 + 0.653219i \(0.226582\pi\)
\(98\) 1.73932 + 0.0949204i 0.175697 + 0.00958841i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 567.2.bd.a.17.10 132
3.2 odd 2 189.2.bd.a.185.13 yes 132
7.5 odd 6 567.2.ba.a.341.13 132
21.5 even 6 189.2.ba.a.131.10 yes 132
27.7 even 9 189.2.ba.a.101.10 ✓ 132
27.20 odd 18 567.2.ba.a.143.13 132
189.47 even 18 inner 567.2.bd.a.467.10 132
189.61 odd 18 189.2.bd.a.47.13 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.10 ✓ 132 27.7 even 9
189.2.ba.a.131.10 yes 132 21.5 even 6
189.2.bd.a.47.13 yes 132 189.61 odd 18
189.2.bd.a.185.13 yes 132 3.2 odd 2
567.2.ba.a.143.13 132 27.20 odd 18
567.2.ba.a.341.13 132 7.5 odd 6
567.2.bd.a.17.10 132 1.1 even 1 trivial
567.2.bd.a.467.10 132 189.47 even 18 inner